On the difference between FOPT and CIPT for hadronic tau decays

Abstract

In this article we review the results of our recent work on the difference between the Borel representations of $$\tau $$ τ hadronic spectral function moments obtained with the CIPT and FOPT methods. For the presentation of the theoretical results we focus on the large-$$\beta _0$$ β 0 approximation, where all expressions can be written down in closed form, and we comment on the generalization to full QCD. The results may explain the discrepancy in the behavior of the FOPT and CIPT series that has been the topic of intense discussions in previous literature and which represents a major part of the theoretical uncertainties in current strong coupling determinations from hadronic $$\tau $$ τ decays. The findings also imply that the OPE corrections for FOPT and CIPT differ and that the OPE corrections for CIPT do not have standard form.

Reconciling the Contour-Improved and Fixed-Order Approaches for $τ$ Hadronic Spectral Moments I: Renormalon-Free Gluon Condensate Scheme

“…Applying the new renormalon-free GC scheme to spectral function moments we corroborate the statements and implications of Refs. [21,22]. In particular we show that the discrepancy between the FOPT and CIPT expansion series for GC suppressed spectral function moments can be indeed removed, such that the CIPT expansion remains as a viable expansion method.…”

“…The concept of the asymptotic separation suggested in Refs. [21,22] is particularly important for the GCS spectral function moments. It states that the discrepancy problem for the FOPT and CIPT expansions of δ (0) W (s 0 ) can be caused by the GC renormalon if the value N 4,0 is not strongly suppressed.…”

“…2 In Ref. [21] (see also the review [22]) it was shown at an analytic level that a discrepancy in the behavior of the FOPT and CIPT spectral function moment series can arise from a different infrared (IR) sensitivity inherent to the two expansion methods. To be concrete, it was demonstrated how the asymptotic (i.e.…”

“…In this work we explore from a practical perspective the suggestion made in Refs. [21,22], that the discrepancy between the CIPT and FOPT spectral function moment perturbation series is of IR origin and mainly caused by the IR renormalon associated to the GC. Even though we use analytic results for the asymptotic separation derived in [21,22] for comparison in our numerical analyses, our concrete considerations do not rely on the concept of the asymptotic separation.…”

Renormalons and hyperasymptotics in QCD

“…The dependence on the renormalization scheme is also discussed in [5], the advantages of the so-called C scheme over the standard MS being emphasized. The problem of renormalization scale dependence is investigated in detail in [6], where Hoang and Regner advocate the existence of different Borel representations of the τ hadronic spectral function moments obtained with two popular prescriptions for the renormalization scale, contour-improved perturbation theory (CIPT) and fixed-order perturbation theory (FOPT), which suggests that the associated OPEs differ. Mariño et al discuss in [7] the Bethe ansatz for the free energy in two-dimensional integrable models, proving that the ansatz is verified up to very high orders in the coupling constant for the non-linear sigma model and its supersymmetric extension.…”